In computer graphics, simplifying a polygonal mesh surface~$\mathcal{M}$ into a geometric proxy that maintains close conformity to~$\mathcal{M}$ is crucial, as it can significantly reduce computational demands in various applications. In this paper, we introduce the Implicit Thin Shell~(ITS), a concept designed to implicitly represent the sandwich-walled space surrounding~$\mathcal{M}$, defined as~$\{\textbf{x}\in\mathbb{R}^3|\epsilon_1\leq f(\textbf{x}) \leq \epsilon_2, \epsilon_1< 0, \epsilon_2>0\}$. Here, $f$ is an approximation of the signed distance function~(SDF) of~$\mathcal{M}$, and we aim to minimize the thickness~$\epsilon_2-\epsilon_1$. To achieve a balance between mathematical simplicity and expressive capability in~$f$, we employ a tri-variate tensor-product B-spline to represent~$f$. This representation is coupled with adaptive knot grids that adapt to the inherent shape variations of~$\mathcal{M}$, while restricting~$f$'s basis functions to the first degree. In this manner, the analytical form of~$f$ can be rapidly determined by solving a sparse linear system. Moreover, the process of identifying the extreme values of~$f$ among the infinitely many points on~$\mathcal{M}$ can be simplified to seeking extremes among a finite set of candidate points. By exhausting the candidate points, we find the extreme values~$\epsilon_1<0$ and $\epsilon_2>0$ that minimize the thickness. The constructed ITS is guaranteed to wrap~$\mathcal{M}$ rigorously, without any intersections between the bounding surfaces and~$\mathcal{M}$. ITS offers numerous potential applications thanks to its rigorousness, tightness, expressiveness, and computational efficiency. We demonstrate the efficacy of ITS in rapid inside-outside tests and in mesh simplification through the control of global error.

翻译：在计算机图形学中，将多边形网格表面~$\mathcal{M}$简化为一个保持与~$\mathcal{M}$紧密贴合性的几何代理至关重要，因为它能显著降低各类应用中的计算需求。本文提出了隐式薄壳~(ITS)的概念，旨在隐式地表征包围~$\mathcal{M}$的夹层壁空间，其定义为~$\{\textbf{x}\in\mathbb{R}^3|\epsilon_1\leq f(\textbf{x}) \leq \epsilon_2, \epsilon_1< 0, \epsilon_2>0\}$。其中，$f$是~$\mathcal{M}$的有符号距离函数~(SDF)的一个近似，我们的目标是最小化厚度~$\epsilon_2-\epsilon_1$。为了在$f$的数学简洁性与表达能力之间取得平衡，我们采用三元张量积B样条来表示$f$。该表示与自适应节点网格相结合，以适应~$\mathcal{M}$固有的形状变化，同时将$f$的基函数限制为一阶。通过这种方式，$f$的解析形式可以通过求解一个稀疏线性系统快速确定。此外，在~$\mathcal{M}$上无穷多个点中寻找$f$极值的过程，可以简化为在有限候选点集中寻找极值。通过穷举候选点，我们找到使厚度最小化的极值~$\epsilon_1<0$和$\epsilon_2>0$。所构建的ITS保证严格包裹~$\mathcal{M}$，其边界曲面与~$\mathcal{M}$之间不存在任何相交。得益于其严格性、紧致性、表达能力和计算效率，ITS具有许多潜在应用。我们通过快速内外测试以及通过控制全局误差进行网格简化，展示了ITS的有效性。